He said Dorchester Collection hotels in the US, UK, France and Italy, which are owned by the Brunei Investment Agency, should be avoided by those who oppose the measures.
Brunei, on the island of Borneo, is ruled by Sultan Hassanal Bolkiah and has grown rich on oil and gas exports.
The Sultan owns the Brunei Investment Agency, which counts some of the world's top hotels in its portfolio, including the Dorchester in London and the Beverly Hills Hotel in Los Angeles.
"I've stayed at many of them," Mr Clooney wrote, "because I hadn't done my homework and didn't know who owned them.
"Every single time we stay at or take meetings at or dine at any of these nine hotels we are putting money directly into the pockets of men who choose to stone and whip to death their own citizens," he added.
Other public figures have also announced they are boycotting the Dorchester Collection.
Filmmaker Dustin Lance Black wrote on Twitter: "If you continue to stay at or frequent the Beverley Hills Hotel, you are guilty of financially supporting these murderers."
BBC world affairs editor John Simpson also confirmed that he wouldn't be visiting hotels owned by the group.
I love the Dorchester Hotel in London, & once had dinner there with the wonderful Martha Gellhorn. But I couldn’t set foot there while, back home in Brunei, its owner introduces stoning to death for adultery & gay sex.